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The Golden R atio. By Rachel Lewis. adapted from http://www.geom.uiuc.edu/~demo5337/s97b/discover.htm. Goal. Students will calculate the Golden Ratio and discover where it exists in the world around them.

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The golden r atio

TheGoldenRatio

By Rachel Lewis

adapted from http://www.geom.uiuc.edu/~demo5337/s97b/discover.htm


The golden r atio
Goal

Students will calculate the Golden Ratio and discover where it exists in the world around them.

  • Given a ruler and various rectangles found in the classroom, students will measure the length and width of each rectangle

  • Using a calculator, students will be able to calculate the ratio of length to width for each rectangle.

  • Using a calculator, students will find the average of their results to estimate the Golden Ratio.

Objectives


Materials
Materials

  • Golden Ratio worksheet

  • Ruler

  • Any rectangular index card

  • Calculator

  • Pencil


Procedure
Procedure

From this picture, students are asked to measure the length and width of their favorite rectangle in centimeters and to record these values in the table in their packets.


The golden r atio

Then measure the:

  • length and width of the index card

  • length and width of the worksheet

  • distance from the knuckle on the back of your hand to the next knuckle (use as length), and then from the second to knuckle to the next (use as width)


The golden r atio

Students are then told to walk around the classroom and find various rectangles to measure. All data is to be recorded in the same table.


Sample table
Sample Table various rectangles to measure. All data is to be recorded in the same table.


Find the ratio
Find the Ratio various rectangles to measure. All data is to be recorded in the same table.

Students are then asked to calculate the ratio of length to width for each rectangle the measure using the formula:


Sample ratios
Sample Ratios various rectangles to measure. All data is to be recorded in the same table.

  • Index Card

    =

  • Worksheet

    =

    *These values should be the same for each student


Complete sample table
Complete Sample Table various rectangles to measure. All data is to be recorded in the same table.


Questions
Questions various rectangles to measure. All data is to be recorded in the same table.

  • What do you notice about your ratios?

  • Take the average of the 8 ratios you found. Record this number on the chart on the board.

  • Do you think this number would change if you measured in inches instead of centimeters?

  • Measure this worksheet in inches and find the ratio. What do you notice?

  • Find the average of the values on the board.


The golden ratio
The Golden Ratio various rectangles to measure. All data is to be recorded in the same table.

The number you have calculated should be close to 1.61803. This is called the Golden Ratio. Remember the Fibonnaci sequence we studied before? Well you will notice that if we find the ratio of consecutive numbers…

2/1 = 2.0 3/2 = 1.5 5/3 = 1.67

8/5 = 1.6 13/8 = 1.625 21/13 = 1.615

34/21 = 1.619 55/34 = 1.618 89/55 = 1.618

the result gets closer and closer to the Golden Ratio! The number first got its fame in Ancient Greece when mathematicians noticed how frequently it appeared in geometry. This ratio is said to be used in architecture from the Parthenon in Greece to the Great Mosque of Kairoun in Tunisia. Leonardo DaVinci’s famous drawing to the left shows a man drawn within a pentagon, suggests that the Golden Ratio exists in the human form.


Some other thoughts
Some Other Thoughts… various rectangles to measure. All data is to be recorded in the same table.

  • Have students measure the distance from their shoulder to elbow and elbow to wrist.

  • Give them a picture of the Parthenon and see if they can find Gold Rectangles

  • Research other places where the Golden Ratio is apparent (art, architecture, etc…)